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Congestion bounds via Laplacian eigenvalues and their application to tensor networks with arbitrary geometry

Sayan Mukherjee, Shinichiro Akiyama

TL;DR

This paper connects tensor-network contraction memory to graph spectral properties by bounding vertex congestion in terms of Laplacian eigenvalues: for any embedding, ${\mathrm cng}(G)$ satisfies $\frac{2\lambda_2(G)}{9}n \le {\mathrm cng}(G) \le \frac{\lambda_n(G)}{4}n$, and there exist embeddings achieving tighter, spectrum-informed bounds. It extends these results to the normalized Laplacian and derives corollaries translating congestion bounds into contraction memory lower/upper bounds and treewidth estimates, thereby linking spectral graph theory to contraction complexity on arbitrary geometries. The authors validate their bounds numerically across hypercubes, lattices, random graphs, and quantum-circuit tensor networks, and show hierarchical spectral clustering as an effective heuristic for low-congestion embeddings on sparse graphs. The work provides mathematical groundwork for predicting and designing contraction strategies with memory guarantees in general tensor-network settings, with potential impact on quantum circuit simulation and inference on complex networks.

Abstract

Embedding the vertices of arbitrary graphs into trees while minimizing some measure of overlap is an important problem with applications in computer science and physics. In this work, we consider the problem of bijectively embedding the vertices of an $n$-vertex graph $G$ into the leaves of an $n$-leaf rooted binary tree $\mathcal{B}$. The congestion of such an embedding is given by the largest size of the cut induced by the two components obtained by deleting any vertex of $\mathcal{B}$. The congestion $\mathrm{cng}(G)$ is defined as the minimum congestion obtained by any embedding. We show that $λ_2(G)\cdot 2n/9\le \mathrm{cng} (G)\le λ_n(G)\cdot 2n/9$, where $0=λ_1(G)\le \cdots \le λ_n(G)$ are the Laplacian eigenvalues of $G$. We also provide a contraction heuristic given by hierarchically spectral clustering the original graph, which we numerically find to be effective in finding low congestion embeddings for sparse graphs. We numerically compare our congestion bounds on different families of graphs with regular structure (hypercubes and lattices), random graphs, and tensor network representations of quantum circuits. Our results imply lower and upper bounds on the memory complexity of tensor network contraction in terms of the underlying graph.

Congestion bounds via Laplacian eigenvalues and their application to tensor networks with arbitrary geometry

TL;DR

This paper connects tensor-network contraction memory to graph spectral properties by bounding vertex congestion in terms of Laplacian eigenvalues: for any embedding, satisfies , and there exist embeddings achieving tighter, spectrum-informed bounds. It extends these results to the normalized Laplacian and derives corollaries translating congestion bounds into contraction memory lower/upper bounds and treewidth estimates, thereby linking spectral graph theory to contraction complexity on arbitrary geometries. The authors validate their bounds numerically across hypercubes, lattices, random graphs, and quantum-circuit tensor networks, and show hierarchical spectral clustering as an effective heuristic for low-congestion embeddings on sparse graphs. The work provides mathematical groundwork for predicting and designing contraction strategies with memory guarantees in general tensor-network settings, with potential impact on quantum circuit simulation and inference on complex networks.

Abstract

Embedding the vertices of arbitrary graphs into trees while minimizing some measure of overlap is an important problem with applications in computer science and physics. In this work, we consider the problem of bijectively embedding the vertices of an -vertex graph into the leaves of an -leaf rooted binary tree . The congestion of such an embedding is given by the largest size of the cut induced by the two components obtained by deleting any vertex of . The congestion is defined as the minimum congestion obtained by any embedding. We show that , where are the Laplacian eigenvalues of . We also provide a contraction heuristic given by hierarchically spectral clustering the original graph, which we numerically find to be effective in finding low congestion embeddings for sparse graphs. We numerically compare our congestion bounds on different families of graphs with regular structure (hypercubes and lattices), random graphs, and tensor network representations of quantum circuits. Our results imply lower and upper bounds on the memory complexity of tensor network contraction in terms of the underlying graph.

Paper Structure

This paper contains 19 sections, 7 theorems, 23 equations, 7 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Main Theorem
  3. Corollaries
  4. Organization
  5. Background and Related works
  6. Graph embedding
  7. Carving width
  8. Treewidth and Laplacian eigenvalues
  9. Sparse cuts in graphs and spectral clustering
  10. Vertex congestion and Laplacian Eigenvalues
  11. Proof of Theorem \ref{['thm:MAINTHM']}
  12. Normalized Laplacian
  13. Vertex congestion of special graph classes
  14. Hypercubes
  15. Non-periodic and Periodic Lattices
  16. ...and 4 more sections

Key Result

theorem 1

For any graph $G$ and embedding $\pi:V(G)\to\ell(\mathcal{B})$, the following holds. Moreover, there are embeddings $\pi_i:V(G)\to\ell(\mathcal{B}_i)$ such that

Figures (7)

  • Figure 1: (Left): A tensor network $(G,T)$ on $6$ nodes where each bond of $T$ has dimension $2$. (Right): The rooted binary tree $\mathcal{B}$ representing a contraction order. A node $S$ of $\mathcal{B}$ represents an intermediate tensor encountered during the contraction procedure and has rank $|\partial S|$, which is indicated at the top of each node. The vertex congestion of this embedding is $4$.
  • Figure 2: Comparison of the different bounds for: (a) the non-periodic lattice $P_m\square P_n$ and (b) periodic lattice $C_m\square C_n$ for $m=5$ and $5\le n \le 20$; (c) non-periodic square lattice and (d) periodic square lattice $P_k\square P_k$ for $2\le k \le 20$. The light green lines represent the lower bound of $2mn\lambda_2(G)/9$ in (\ref{['eq:experiments-lattice-mainthm-lower']}), brown the upper bound of Theorem \ref{['thm:MAINTHM']} in (\ref{['eq:experiments-lattice-mainthm-upper']}), green and red the lower and upper treewidth bounds of (\ref{['eq:experiments-lattice-treewidth']}).
  • Figure 3: Congestion in $50$ instances of $\mathcal{G}(n,d)$, shown as mean values with error bars for $\pm1$ sample standard deviation, for (a) $d=3$ and (b) $d=4$, for $n=10$ to $n=24$. Brown data represent the upper bound of $n\lambda_n(G)/4$, maroon dots the theoretical upper bound of Theorem \ref{['thm:MAINTHM']}, and light green dots the lower bound of $2n\lambda_2(G)/9$. Insets compare the performance of hierarchical clustering (Algorithm \ref{['alg:HSC']}), Hyper-Greedy, Cotengra-Auto, and Hyper-Opt.
  • Figure 4: Congestion in $50$ instances of $\mathcal{G}_{n,p}$, shown as mean values with error bars for $\pm1$ sample standard deviation, for $p=0.15$ (subplot (a)) and $p=0.2$ (subplot (b)) for $n=10$ to $n=24$. Brown data represent the upper bound of $4m\mu_n(G)/9$ ((\ref{['eq:normalized-mainthm-3']}), maroon dots the theoretical upper bound of Theorem \ref{['thm:normalizedMAINTHM']}, and light green dots the lower bound of $4m\mu_2(G)/9$ (\ref{['eq:normalized-mainthm-1']}). Insets compare the performance of hierarchical clustering (Algorithm \ref{['alg:HSC']}), Hyper-Greedy, Cotengra-Auto, and Hyper-Opt.
  • Figure 5: Congestions obtained by HSC (Algorithm \ref{['alg:HSC']}), Hyper-Greedy, Cotengra-Auto, and Hyper-Opt in $50$ instances of $\mathcal{G}_{15,p}$, shown as mean values with error bars for $\pm1$ sample standard deviation, for $p=0.12$ to $0.185$.
  • ...and 2 more figures

Theorems & Definitions (8)

  • Definition 1: $\varepsilon(G)$
  • theorem 1
  • Corollary 1
  • Corollary 2
  • lemma 1: Discrete Cheeger Inequalities LaplacianSurvey-mohar-1991
  • theorem 2
  • lemma 2: Friedman, proofOfAlonsSecondEigenvalueConjecture-Friedman-2003
  • lemma 3: Chung and Radcliffe GnpSpectrum-chung-radcliffe-2011, Ding and Tiang GnpSpectrum-ding-tiang-2010